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There are two sets A and B with equal geometric mean and arithmetic mean. Each element of both sets is odd integer greater than 1. A = B ? Order of elements isn't important.

Dmitriy Finozhenok
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  • @Ikki By my definition of a set they are the same – user127.0.0.1 Jan 14 '14 at 14:41
  • If the sets $A,B$ can be of different sizes, this is not immediately obvious. – coffeemath Jan 14 '14 at 18:43
  • It should be made clear: If $A(S),G(S)$ are the arithmetic mean and geometric mean of a set $S$, is one looking for unequal sets $X,Y$ for which the two equations $A(X)=A(Y)$ and $G(X)=G(Y)$ should hold? Or on the other hand are all four of $A(X),A(Y),G(X),G(Y)$ to be equal? Probably the former, as if all four are equal then each "set" has only one element and it's trivial. Also make clear whether the two sets are to have the same size. – coffeemath Jan 14 '14 at 21:12

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Is there another constraint about cardinality? As it stands an obvious counterexample is $A = \{3\}, B = \{3,3\}$.

Eddie E.
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    But $A=B$ in your example :D – dani_s Jan 14 '14 at 15:01
  • The cardinality of these $A,B$ is both one, and in fact $A=B$ also. So this is not a counterexample to the question of the OP. If the question were in terms of multisets, OK, but it wasn't. – coffeemath Jan 15 '14 at 14:42